Noncommutative Spacetime and the Generalised Uncertainty Principle from Worldline Non-Injectivity: A Geometric Derivation of -Minkowski and the GUP

We derive the κ‑Minkowski noncommutative spacetime and the Generalised Uncertainty Principle (GUP) from a single classical geometric principle: worldline non‑injectivity. In standard relativity, a timelike worldline \ (X^ () \) is assumed to intersect each constant‑time hypersurface \ (ₜ\) exactly once (injectivity). This assumption is never stated as an axiom, yet it underpins the uniqueness of Lorentz transformations and the usual formulation of quantum fields. When the Lorentz factor exceeds a critical threshold \ (> ₂ₑ₈ₓ\), injectivity fails: the same worldline intersects \ (ₜ\) in \ (N>1\) distinct spatial points. This is **worldline non‑injectivity** – a purely kinematic consequence of special relativity, not an additional postulate. The \ (N\) simultaneous intersections generate a discrete Hilbert space of topological sheets \ (H ₒ₇₄₄ₓₒ = ² (ZN) \). On this space, the natural shift and phase operators satisfy the Weyl relation \ (UV=e^2 i/NVU\). Translating these operators into physical coordinates via the Extended Lorentz Transformations (ELT) of the multi‑sheet framework yields two fundamental results with no free parameters: Generalised Uncertainty Principle (GUP) > \> [X, P = i (1 - 2N) > = i (1 - 2\, p²mP² c²), > \] where the deformation coefficient \ (= 2\) is fixed by the fold stability condition \ (_=2\) (the same condition that defines Planck’s constant from inter‑sheet interference). → The minimum measurable length is \ (X_ = 2\, P 2. 51\, P\), a precise prediction that distinguishes this model from other GUP proposals. 2. κ‑Minkowski noncommutative spacetime> \> [X⁰, X¹ = iX¹, = ₂ₑ₈ₓc, > \] with the deformation parameter \ (\) fixed by the cosmological UV cutoff \ (= P/LH\) (derived from the observed cosmological constant). In the Planck‑scale limit \ (= P\), one obtains \ (= mP c/ = 1/P\) – the Planck mass emerges from geometry, not from an external input. 3. κ‑Poincaré algebra The ELT generators reproduce the standard Poincaré algebra at leading order, with sheet‑dependent corrections of order \ (1/²\) that match the expected form of the κ‑Poincaré Hopf algebra. All deformation parameters are **derived, not postulated**. The only external input is the observed value of the cosmological constant, which sets the scale \ (= P/LH\). The Planck length \ (P\) itself follows from the critical Lorentz factor and the Hubble radius. This work completes the unification of holography, quantum mechanics, classical electrodynamics, thermodynamics, quantum statistics, and noncommutative geometry under the single principle of worldline non‑injectivity. The universal cancellation identity \ (N () ^d-2=O (1) \) now operates at eight distinct levels of physical theory, from RT entropy to the GUP. This manuscript is current in Official Peer Review. Not final version. Copyright©2026 Alex De Giuseppe. All rights reserved. This work is protected by copyright. Any form of plagiarism, unauthorized reproduction, or misappropriation of ideas, mathematically results, or text without proper citation constitutes a violation of academic and intellectual property standards and common laws. No commercial use, adaptation, or derivative works are permitted without explicit written permission from the author. For correspondence, citations, collaboration inquiries, or feedback please contact: degiuseppealex@gmail. com The hash files that determine ownership have been created